# Singular Solutions of Elliptic Equations with Iterated Exponentials

@article{Ghergu2019SingularSO, title={Singular Solutions of Elliptic Equations with Iterated Exponentials}, author={Marius Ghergu and Olivier Goubet}, journal={The Journal of Geometric Analysis}, year={2019}, volume={30}, pages={1755-1773} }

We construct positive singular solutions for the problem $$-\Delta u=\lambda \exp (e^u)$$ - Δ u = λ exp ( e u ) in $$B_1\subset {\mathbb {R}}^n$$ B 1 ⊂ R n ( $$n\ge 3$$ n ≥ 3 ), $$u=0$$ u = 0 on $$\partial B_1$$ ∂ B 1 , having a prescribed behaviour around the origin. Our study extends the one in Miyamoto (J Differ Equ 264:2684–2707, 2018) for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.

## 3 Citations

Singular solutions to $k$-Hessian equations with fast-growing nonlinearities

- Mathematics
- 2021

We study a class of elliptic problems, involving a k-Hessian and a very fast-growing nonlinearity, on a unit ball. We prove the existence of a radial singular solution and obtain its exact asymptotic…

The structure of radial solutions for a general MEMS model

- Mathematics
- 2020

We investigate the structure of radial solutions corresponding to the equation \[ \Delta u=\frac{1}{f(u)}\ \ \textrm{in}\ B_{r_0}\subset\mathbb{R}^N,\ N\ge 3,\ r_0>0, \] where $f\in C[0,\infty)\cap…

Radial single point rupture solutions for a general MEMS model

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We study the initial value problem $$ \begin{cases} r^{-(\gamma-1)}\left(r^{\alpha}|u'|^{\beta-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0 0 & \textrm{for}\ 0 \alpha>\beta\geq 1$ and $f\in C[0,\bar…

## References

SHOWING 1-10 OF 25 REFERENCES

Perturbing singular solutions of the Gelfand problem

- Mathematics
- 2007

he equation $-\Delta u = \lambda e^u$ posed in the unit ball $B \subseteq \R^N$, with homogeneous Dirichlet condition $u|_{\partial B} = 0$, has the singular solution $U=\log\frac1{|x|^2}$ when…

Partial regularity for a Liouville system

- Mathematics
- 2013

Let $\Omega\subset\mathbb{R}^n$ be a bounded smooth open set.
We prove that the singular set of any extremal solution of the system
\begin{equation*}
-\Delta u=\mu e^v , \quad
- \Delta v=\lambda…

The Gel’fand Problem for the Biharmonic Operator

- Mathematics
- 2013

We study stable and finite Morse index solutions of the equation $${\Delta^2 u = {e}^{u}}$$. If the equation is posed in $${\mathbb{R}^N}$$, we classify radial stable solutions. We then construct…

A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We consider the following semilinear elliptic equation: where B 1 is the unit ball in ℝ d , d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λ…

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

- Mathematics
- 2006

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

- Mathematics
- 2018

The Dirichlet problem for $-\Delta \varphi= \mathrm{e}^{-\varphi}$ in an infinite sector. Application to plasma equilibria

- Mathematics
- 2014

We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from…

Infinitely many nonradial singular solutions of $\Delta u+e^u=0$ in $\mathbb{R}^N\backslash\{0\}$, $4\le N\le 10$

- Mathematics
- 2017

We construct countably infinitely many nonradial singular solutions of the problem \[ \Delta u+e^u=0\ \ \textrm{in}\ \ \mathbb{R}^N\backslash\{0\},\ \ 4\le N\le 10 \] of the form \[…

Solution radiale singulière de −Δu=λeu

- Mathematics
- 1988

We study in R N , N≥3, the radial symmetric solution, singular at the origin, of the problem −Δu=λe u , set either in the whole space, or in a ball with Dirichlet boundary condition. Using phase…

A Liouville-Gelfand Equation for k-Hessian Operators

- Mathematics
- 2004

In this paper we establish existence and multiplicity results for a class of fully nonlinear elliptic equations of k-Hessian type with exponential nonlinearity. In particular, we characterize the…